30.3 sf_5_point

This section was contributed by Charles Goldberg. The five point form approximates averages of contributions to the corner points of the nine point operator to produce a five point operator that has coefficients only in the center, north, east, south, and west places in the coefficient matrix for inverting the external mode elliptic equation. This option can be used with option stream_function but is not appropriate for options rigid_lid_surface_pressure or implicit_free_surface because of energy leakage.

In Equation (30.9), corresponding to the first bracket of Equation (29.2), all 32 contributions to the elliptic operator are calculated as in the 9 point operator, except that in the first group, i.e., in the terms originating in a second difference in the $\delta_\phi$ direction, the coefficient of the northeast variable $\Delta\phi_{i,jrow,1,1}$ is averaged with the coefficient of the northwest variable $\Delta\phi_{i,jrow,-1,1}$, and both are applied to the northern variable $\Delta\phi_{i,jrow,0,1}$. This averaging centers the coefficient on the northern edge of the central T cell, so that continuous derivatives in the $\phi$ direction are well approximated. Similarly, the southeast and southwest contributions from the first group are applied to the southern variable $\Delta\phi_{i,jrow,0,-1}$. The second group of terms originate in a second difference in the $\delta_\lambda$ direction, and here the northeastern and southeastern contributions are averaged and applied to the eastern variable $\Delta\phi_{i,jrow,1,0}$, and the northwestern and southwestern contributions are averaged and applied to the western variable $\Delta\phi_{i,jrow,-1,0}$, giving good approximations to the continuous derivatives in the $\lambda$ direction. The resulting summations are

 

$$\hskip-2em\sum_{\ip=0}^{1}\sum_{\jp=0}^{1} \sum_{\ipp=-1}^{0} \sum_{\jpp=-1}^{0} \hskip-2em\left[ cddyu_{\ip,\jp}\cdot cddyt_{\ipp,\jpp}\cdot \fra... ...jpp}\cdot dyu_{jrow+\jpp}\cdot2\Delta\tau} \right] \Delta\psi_{i,jrow+\jp+\jpp}$$

$$\hskip-2em + \sum_{\ip=0}^{1}\sum_{\jp=0}^{1} \sum_{\ipp=-1}^{0} \sum_{\jpp=-1}^{0} \hskip-2em\left[ cddxu_{\ip,\jp}\cdot cddxt_{\ipp,\jpp}\cdot \fra... ...ot \cos\phi^U_{jrow+\jpp}\cdot2\Delta\tau} \right] \Delta\psi_{i+\ip+\ipp,jrow}$$

Bryan (1969) uses a five point approximation to $(\frac{1}{H}\cdot \nabla\psi_t)$ which as implemented in MOM 1 takes the form

$${1\over \cstj\dxti\; \dytj} \hskip-2em \left[ \dxti\cdot \dytj \cdot\dely\left( \frac{\csujm}{\overline{\himjm}^\lambda\cdot\dyujm}\; \dyujm \cdot\dely\dpsis \right)\right.$$

$$\hskip-4em +\left. \dytj\cdot dxt_i \cdot \delx\left( \frac{1}{\o... ...ne{\himjm}^\phi \cdot \cstj\; \dxuim}\; \dxuim \cdot\delx\dpsiw \right) \right]$$

In this notation, the five point approximation used in MOM takes the form

$${1\over 2\Delta\tau} \cdot \hskip-1em\left[ \dytj \cdot\dely\left( \overline{ \left(\frac{\d... ...mjm\cdot \dyujm}\right) }^\lambda \cdot \dyujm \cdot\dely\dpsis \right) \right.$$

$$+ \left.dxt_i \cdot \delx\left( \overline{ \left(\frac{\dyujm}{\h... ...sujm \cdot \dxuim}\right) }^\phi \cdot \dxuim \cdot \delx\dpsiw \right) \right]$$

For comparison with MOM, Bryan's form must be multiplied by $(\cstj\dxti\dytj) /(2\Delta\tau)$. The principal differences between these two forms arise from Bryan's use of averages of reciprocals of the form $${2\over H_{i,jrow} + H_{i-1,jrow}}$$ where MOM uses $${1\over 2}({1\over H_{i,jrow}} + {1\over H_{i-1,jrow}})$$. These differ by a factor of two in the second order term in H_i,jrow - H_i-1,jrow. Other differences arise on a nonuniform grid.

The second bracket, the implicit Coriolis terms, may be seen to already be in five point form because each corner coefficient consists of two terms of equal magnitude, but opposite sign.

 

$$\sum_{\ip=0}^{1}\sum_{\jp=0}^{1} \sum_{\ipp=-1}^{0} \sum_{\jpp=-1}^{0} \hskip-2em\left[ -cddxu_{\ip,\jp}\cdot cddyt_{\ipp,\jpp}\cdot \frac{-\tilde{f}_{jrow+\jpp}}{H_{i+\ipp,jrow+\jpp}} \right.$$

$$\hskip-1em\left. -cddyu_{\ip,\jp}\cdot cddxt_{\ipp,\jpp}\cdot \fr... ...jrow+\jpp}}{H_{i+\ipp,jrow+\jpp}} \right] \Delta\psi_{i+\ip+\ipp,jrow+\jp+\jpp}$$ (30.12)

The right hand side of Equation (29.14) also remains the same in the five point equations as in the nine point equations.

 

$$forc_{i,jrow,} = \sum_{\ip=0}^{1}\sum_{\jp=0}^{1} \hskip-2em\left[-cddyt_{\ip,\jp}\cdot zu_{i+\ip,jrow+\jp,1} \cdot dxu_{i+\ip}\cdot\cos\phi^U_{jrow+\jp} \right.$$

$$\hskip-1em\left.+cddxt_{\ip,\jp}\cdot zu_{i+\ip,jrow+\jp,2} \cdot dyu_{jrow+\jp} \right]$$ (30.13)

Although the five point operator approximates the continuous differential equation (29.2) well, its solutions are not exact solutions of the finite difference momentum equations (29.3) and (29.4). Moreover, the time saved by calculating a five point operator instead of a nine point operator in two places per iteration in a conjugate gradient solver is not large, and the nine point equations usually take fewer iterations to converge.